Mathematical Programming

, Volume 159, Issue 1, pp 585–605

On sublinear inequalities for mixed integer conic programs

Short Communication Series A

DOI: 10.1007/s10107-015-0968-0

Cite this article as:
Kılınç-Karzan, F. & Steffy, D.E. Math. Program. (2016) 159: 585. doi:10.1007/s10107-015-0968-0

Abstract

This paper studies \(\mathcal {K}\)-sublinear inequalities, a class of inequalities with strong relations to \(\mathcal {K}\)-minimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of \(\mathcal {K}\)-sublinear inequalities. That is, we show that when \(\mathcal {K}\) is the nonnegative orthant or the second-order cone, \(\mathcal {K}\)-sublinear inequalities together with the original conic constraint are always sufficient for the closed convex hull description of the associated disjunctive conic set. When \(\mathcal {K}\) is the nonnegative orthant, \(\mathcal {K}\)-sublinear inequalities are tightly connected to functions that generate cuts—so called cut-generating functions. In particular, we introduce the concept of relaxed cut-generating functions and show that each \({\mathbb {R}}^n_+\)-sublinear inequality is generated by one of these. We then relate the relaxed cut-generating functions to the usual ones studied in the literature. Recently, under a structural assumption, Cornuéjols, Wolsey and Yıldız established the sufficiency of cut-generating functions in terms of generating all nontrivial valid inequalities of disjunctive sets where the underlying cone is nonnegative orthant. We provide an alternate and straightforward proof of this result under the same assumption as a consequence of the sufficiency of \(\mathbb {R}^n_+\)-sublinear inequalities and their connection with relaxed cut-generating functions.

Keywords

Valid inequalities Sublinear inequalities Cut-generating functions Mixed integer conic programming 

Mathematics Subject Classification

Primary 90C11 Secondary 52A41 90C26 

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Mathematics and StatisticsOakland UniversityRochesterUSA

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